This invention relates to carrier recovery in digital receivers. Digital wireless communication systems, such as broadcast, wireless LAN, and wide area mobile systems, use quadrature amplitude modulation (“QAM”). QAM is also used in both North American and European digital cable television standards. QAM uses quadrature-carrier multiplexing to allow two double-side-band suppressed-carrier signals modulated by independent messages to occupy the same channel bandwidth.
A QAM modulator 100 is illustrated in FIG. 1, and an isolated, transmitted QAM pulse from the modulator 100 is given below in EQN. 1.sm(t)=dR,mq(t)cos(2πfct)−dI,mq(t)sin(2πfct)=Re[dmq(t)ej2πfct]  EQN. 1where fc is a modulating carrier frequency; q(t) 105 is a pulse shaping filter; dR,m and dI,m represent a real or in-phase component and an imaginary or quadrature component, respectively, of a complex QAM symbol; and m=1 . . . M indexes a 2-dimensional QAM constellation of cardinality M. A continuous series of transmitted QAM pulses, sm(t), passes through a noisy multipath channel at a rate of fs=1/Ts, where fs is the symbol rate and Ts is the symbol duration time. Thus, the received signal at the input to a QAM demodulator 110 (see FIG. 2) of a receiver is given by r(t)=sm(t)*c(t)+ν(t), where * denotes convolution, c(t) is the channel impulse response, and ν(t) is additive white Gaussian noise. The received signal, r(t), is given below in EQN. 2.
                              r          ⁡                      (            t            )                          =                              Re            ⁢                          {                                                ⅇ                                                            j                      ⁢                                                                                          ⁢                      2                      ⁢                                              π                        ⁡                                                  (                                                                                    f                              LO                                                        +                                                          f                              o                                                                                )                                                                    ⁢                      t                                        +                                          θ                      o                                                                      ⁢                                                      ∑                                          n                      =                                              -                        ∞                                                                                    +                      ∞                                                        ⁢                                                                          ⁢                                                            [                                                                        d                          ⁡                                                      [                            n                            ]                                                                          *                                                  q                          ⁡                                                      (                            t                            )                                                                                              ]                                        ⁢                                          c                      ⁡                                              (                                                  t                          -                                                      nT                            s                                                                          )                                                                                                        }                                +                      υ            ⁡                          (              t              )                                                          EQN        .                                  ⁢        2            where d[n] is the complex transmitted symbol, and fo and θo are the frequency and phase offsets, respectively, of a local oscillator with respect to the modulator 100. The frequency of the local oscillator is given as fLO=fc−fo.
For descriptive purposes, the receiver is assumed to have perfect symbol timing recovery. As such, sampling r(t) at kTs, where k is an integer timing index, results in EQN. 3.
                              r          ⁡                      (                          kT              s                        )                          =                              Re            ⁢                          {                                                ⅇ                                                                                    j2π                        ⁡                                                  (                                                                                    f                              LO                                                        +                                                          f                              o                                                                                )                                                                    ⁢                                              kT                        s                                                              +                                          θ                      o                                                                      ⁢                                                      ∑                                          n                      =                                              -                        ∞                                                                                    +                      ∞                                                        ⁢                                                                          ⁢                                                            [                                                                        d                          ⁡                                                      [                            n                            ]                                                                          *                                                  q                          ⁡                                                      (                                                          kT                              s                                                        )                                                                                              ]                                        ⁢                                          c                      ⁡                                              (                                                                              kT                            s                                                    -                                                      nT                            s                                                                          )                                                                                                        }                                +                      υ            ⁡                          (                              kT                s                            )                                                          EQN        .                                  ⁢        3            After the received signal is matched filtered and demodulated, the signal input to an equalizer 115 is given below by EQN. 4.
                              x          ⁡                      (                          kT              s                        )                          =                              x            ⁡                          [              k              ]                                =                                                    ⅇ                                                      j                    ⁢                                                                                  ⁢                    2                    ⁢                    π                    ⁢                                                                                  ⁢                                          f                      o                                        ⁢                                          kT                      s                                                        +                                      θ                    o                                                              ⁢                                                ∑                                      n                    =                                          -                      ∞                                                                            +                    ∞                                                  ⁢                                                                  ⁢                                                      d                    ⁡                                          [                      n                      ]                                                        ⁢                                      c                    ⁡                                          [                                              k                        -                        n                                            ]                                                                                            +                                          υ                ′                            ⁡                              [                k                ]                                                                        EQN        .                                  ⁢        4            where ν′[k] is complex filtered noise, and inter-symbol interference (“ISI”) present in x(k) is due only to the channel impulse response c(k).
Following demodulation and assuming perfect equalization, a near-baseband complex sequence, y[k], shown in EQN. 5 is output from the equalizer 115.y[k]=d[k]ej2πfokTs+θ0+ν′[k]  EQN. 5As such, the recovered near-baseband sequence represents a transmitted constellation rotating at a frequency, fo, and having a phase offset, θo. For the receiver to reliably recover the transmitted complex QAM symbol (i.e., dR,M and dI,M), using, for example, a two-dimensional slicer, the receiver must remove the frequency offset, fo, that causes the constellation to rotate. The receiver must also remove the phase offset, θo, which otherwise leaves the constellation in a static-rotated position.
An example of a transmitted constellation 200 for 4-QAM (also known as quadrature phase-shift keying (“QPSK”)) prior to modulation is illustrated in FIG. 3, and an example of a received, rotated constellation 205 with a phase offset, θo, is illustrated in FIG. 4. The number of samples required to see a full 360-degree rotation of the constellation is given by fs/fo, and the direction of rotation depends on whether the frequency offset, fo, is positive or negative with respect to the modulating carrier frequency, fc. For example, if a symbol, a, is repeatedly transmitted and fs/fo=−6 and θo=−45°, the first 6 received samples are positioned as illustrated in the constellation 210 of FIG. 5.
Some carrier recovery algorithms (e.g., blind carrier recovery algorithms) extract the carrier frequency and phase information using nonlinear operations. If the frequency offset, fo, is large, the high bandwidth filters used in the receiver introduce a significant amount of noise into the detected phase error signal. The noise limits the frequency-offset pull-in range of a receiver and causes phase jitter in the received signals.
Other carrier recovery algorithms use a blind equalization algorithm in combination with a decision directed (“DD”) carrier recovery algorithm. The blind equalization algorithm is used to equalize a communications channel having phase ambiguity, and the DD carrier recovery algorithm detects the phase error based on the equalizer output and a symbol decision. For example, a transmitted symbol is given as d[k], an output of the equalizer 115 is given as y[k], a frequency offset is given as f, and an initial carrier phase offset is given as θo. If perfect equalization takes place (and ignoring noise), the output of the equalizer 115 is given by EQN. 6.y[k]=d[k]ej2πfokT+θo  EQN. 6The phase error, φ[k], between the equalizer output, y[k], and transmitted symbol, d[k], is given by EQN 7.φ[k]=2πfokT+θo  EQN. 7
A frequency detector can achieve fast acquisition and successful frequency-offset pull-in using a differential error, e[k], between the phase error at a first time, φ[k], and the phase error at a second (delayed) time, φ[k−1], as shown below in EQN 8.e[k]=φ[k]−φ[k−1]=2πfoT  EQN. 8If the differential error, e[k], is positive, a positive frequency offset exists. If the differential error, e[k], is negative, a negative frequency offset exists. Additionally, the magnitude of the differential error, e[k], is proportional to the absolute value of a frequency offset, fo.
The phase error, φ[k], is detected using any of a variety of techniques. One such technique is described below with reference to EQN. 9.{circumflex over (φ)}[k]=CIm(y[k]{circumflex over (d)}*[k])  EQN. 9where C is a constant, Im indicates the imaginary component of a complex number, (●) indicates a conjugate, and {circumflex over (d)}[k] is a two-dimensional sliced symbol decision based on the equalizer output, y[k] EQN. 9 indicates that the decision device affects the phase estimation. Additionally, when using DD carrier recovery techniques, decision boundaries cause a phase jump (e.g., a 90° phase jump). As an illustrative example, FIG. 6 illustrates a QPSK constellation 215 and decision boundaries 220. FIG. 7 illustrates a phase error plot 225 and the phase jumps which occur when the phase error, {circumflex over (φ)}(k), crosses one of the decision boundaries 220. As shown in FIG. 7, the phase error is periodic. As a result of the phase jump, the frequency offset, fo, which is determined using the differential error, is no longer valid. Averaging the differential error results in a value of zero, and therefore, does not provide a correct indication of the direction of the frequency offset, fo. As an additional consequence, as the frequency offset, fo, increases in magnitude, the phase jump is increasingly difficult to identify, because a large frequency offset makes each phase change appear to be a phase jump. Accordingly, conventional carrier recovery techniques fail in the presence of large frequency offsets.